A120257 Triangle of Hankel transforms of certain binomial sums.

1, 2, -1, 3, -6, -1, 4, -20, -20, 1, 5, -50, -175, 70, 1, 6, -105, -980, 1764, 252, -1, 7, -196, -4116, 24696, 19404, -924, -1, 8, -336, -14112, 232848, 731808, -226512, -3432, 1, 9, -540, -41580, 1646568, 16818516, -24293412, -2760615, 12870, 1, 10, -825, -108900, 9343620, 267227532, -1447482465

Offset: 0

Author

Paul Barry, Jun 13 2006

Keywords

• easy
• sign
• tabl

Comments

Row k is the Hankel transform of Sum_{j=0..n} binomial(k+j, j). Absolute value is reversal of A103905. Diagonal and subdiagonals are essentially signed versions of the central coefficients of certain generalized Pascal-Narayana triangles (A007318, A001263, A056939, A056940, A056941).

Examples

			Triangle begins
  1;
  2,   -1;
  3,   -6,     -1;
  4,  -20,    -20,      1;
  5,  -50,   -175,     70,      1;
  6, -105,   -980,   1764,    252,      -1;
  7, -196,  -4116,  24696,  19404,    -924,    -1;
  8, -336, -14112, 232848, 731808, -226512, -3432, 1;
		

Crossrefs

Cf. A120258.

Programs

PARI

T(n, k) = (-1)^((k+1)\2) * prod(j=0, n-k-1, binomial(2*k+2+j, k+1)/binomial(k+1+j, j)); \\ Michel Marcus, Jan 13 2022

Formula

T(n, k) = (cos(Pi*k/2) - sin(Pi*k/2)) * Product_{j=0..n-k-1} C(2k+2+j, k+1)/C(k+1+j, j).